A235154 -id:A235154 - cp's OEIS frontend

A057659 Prime numbers whose square is composed of just two different decimal digits.

5, 7, 11, 109, 173, 81619

Offset: 1

Carlos Rivera, Oct 15 2000

Conjectured to be a finite sequence.

No further terms up to prime(1000000)=15485863. - Harvey P. Dale, Mar 20 2011

			81619^2 = 6661661161.

Related to the Unsolved problem F24, p. 262, of the UPiNT by R. K. Guy

Subsequence of A016069.

Cf. A235154.

filter:= p -> nops(convert(convert(p^2,base,10),set))= 2:
select(filter, [seq(ithprime(i),i=1..10^5)]); # Robert Israel, Feb 11 2025

Select[Prime[Range[10000]],Count[DigitCount[#^2],0] ==8&] (* Harvey P. Dale, Mar 20 2011 *)

select(x->#Set(digits(x^2))==2, primes(10000)) \\ Michel Marcus, Feb 11 2025

A325934 Primes consisting of a single 1 and at least one copy of some other digit.

Original entry on oeis.org

13, 17, 19, 31, 41, 61, 71, 199, 313, 331, 661, 881, 919, 991, 1777, 1999, 2221, 3313, 3331, 4441, 6661, 7177, 7717, 9199, 31333, 33331, 71777, 99991, 199999, 313333, 331333, 333331, 991999, 999199, 3331333, 3333133, 3333313, 3333331, 9999991, 19999999

Offset: 1

Harvey P. Dale, Sep 09 2019

The second Mathematica program below is more complicated than the first but is more efficient. It takes advantage of the observation that any number with a single digit one and one or more copies of another digit from among 2, 4, 5, 6, or 8 can only be prime if the one is the last (least significant) digit. Thus, there is no need to generate or test any permutations of such a number. This means that the program generates and tests only 37.5% as many candidate numbers as the first Mathematica program below. On my laptop computer, in 2019, the first Mathematica program took about 8.2 seconds to compute all terms containing up to 200 digits, whereas the second Mathematica program only took about 6.4 seconds to do the same. - Harvey P. Dale, Sep 20 2019

A further improvement could be made by not testing any permutations of one together with 2, 5, 8, 11, etc. copies of seven, since any such number will have a digital sum of a multiple of three and thus cannot be prime. - Harvey P. Dale, Sep 23 2019

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Subsequence of A208270. Subsequence of A235154.

Select[Flatten[Table[FromDigits/@Permutations[PadRight[{1},n,k]],{n,10},{k,Range[2,9]}]],PrimeQ]//Union
Module[{nn=10,c1,c2},c1=Select[Table[FromDigits[PadLeft[{1},n,k]],{k,{2,4,5,6,8}},{n,2,nn}]//Flatten,PrimeQ];c2=Select[FromDigits/@ Flatten[ Permutations/@Flatten[Table[PadLeft[{1},n,k],{k,{3,7,9}},{n,2,nn}],1],1],PrimeQ];Sort[Flatten[Join[{c1,c2}]]]] (* Harvey P. Dale, Sep 20 2019 *)

lista(nn) = {forprime(p=1, nn, my(d = digits(p)); if ((#Set(d) == 2) && (#select(x->(x==1), d) == 1), print1(p, ", ")););} \\ Michel Marcus, Sep 11 2019

A112750 Smallest prime of the form 7 followed by j copies of the digit k, where j runs through those positive values for which such a prime exists.

Original entry on oeis.org

71, 733, 7333, 79999, 733333, 71111111, 799999999, 79999999999, 79999999999999999999999999, 79999999999999999999999999999999999999999999999999, 733333333333333333333333333333333333333333333333333333, 71111111111111111111111111111111111111111111111111111111

Offset: 1

Amarnath Murthy, Jan 02 2006

For all j > 0, k must be 1, 3, or 9, since a number with --
-- digits 7kk...kk where k is even will be a proper multiple of 2;
-- digits 755...55 will be a proper multiple of 5; and
-- digits 777...77 will be a proper multiple of 7.

			7333 is a term because it is prime and is 7 followed by three copies of 3, and the numbers 7000, 7111, and 7222 are all nonprime.
From _Jon E. Schoenfield_, Feb 23 2021:  (Start)
Terms begin as follows:
   n   j  k  a(n)
  --  --  -  --------------------------------------------------------
   1   1  1  71
   2   2  3  733
   3   3  3  7333
   4   4  9  79999
   5   5  3  733333
   -   6  - (7111111, 7333333, 7999999 are composite)
   6   7  1  71111111
   7   8  9  799999999
   -   9  - (7111111111, 7333333333, 7999999999 are composite)
   8  10  9  79999999999
   -  11  - (711111111111, 733333333333, 799999999999 are composite)
   -  12  - (all composite)
   -  13  - (all composite)
      ...
   9  25  9  79999999999999999999999999
      ...
  10  49  9  79999999999999999999999999999999999999999999999999
      ...
  11  53  3  733333333333333333333333333333333333333333333333333333
  12  55  1  71111111111111111111111111111111111111111111111111111111
(End)

Subsequence of A090155 and hence A235154.

SelectFirst[#,PrimeQ]&/@Table[FromDigits[PadRight[{7},n,p]],{n,2,60},{p,{1,3,9}}]/.Missing["NotFound"]->Nothing (* Harvey P. Dale, Apr 19 2021 *)

More terms added by Harvey P. Dale, Jan 24 2010
Name corrected (using a suggestion from Felix Fröhlich) and Example edited by Jon E. Schoenfield, May 28 2019
Terms corrected by Jon E. Schoenfield, Feb 23 2021

A242846 Palindromic primes of the form ababa...aba containing only the digits 1 and 3.

Original entry on oeis.org

131, 313, 1313131313131313131313131, 313131313131313131313131313131313131313131313131313

Offset: 1

Felix Fröhlich, May 23 2014

The next two terms both start with 3 and have 83 and 225 digits, respectively. Those are the only other terms with fewer than 352 digits. Cf. A062216.

Cf. A062216, A032758, A048399, A235154.

Module[{nn=60,a,b},a=Table[FromDigits[Join[PadRight[{},2n,{1,3}],{1}]],{n,nn}];b=Table[FromDigits[Join[PadRight[{},2n,{3,1}],{3}]],{n,nn}];Select[Sort[Join[a,b]],PrimeQ]] (* Harvey P. Dale, Sep 07 2020 *)

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A057659 Prime numbers whose square is composed of just two different decimal digits.

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A325934 Primes consisting of a single 1 and at least one copy of some other digit.

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A112750 Smallest prime of the form 7 followed by j copies of the digit k, where j runs through those positive values for which such a prime exists.

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A242846 Palindromic primes of the form ababa...aba containing only the digits 1 and 3.

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Mathematica